THE BSE-PROPERTIES FOR VECTOR-VALUED Lp-ALGEBRAS

Abstract

Let A be a separable Banach algebra, G be a locally compact group and 1 < p < ∞. We first provide a necessary and sufficient condition for which Lp(G, A) is a Banach algebra, under convolution product. Then we characterize the character space of Lp(G, A), in the case where A is commutative and G is abelian. Moreover, we investigate the BSE-property for Lp(G, A) and prove that Lp(G, A) is a BSE-algebra if and only if A is a BSE-algebra and G is finite. Finally, we study the BSE-norm property for Lp(G, A) and show that if Lp(G, A) is a BSE-norm algebra then A is so. We prove the converse of this statement for the case where G is finite and A is a unital BSE-algebra. © Rocky Mountain Mathematics Consortium.